Non-Renewal Month by Month Lease & Moving Out Issue

The apartment we are staying in is remodeling and were told about the new owners in November. The new owners gave us a notice on December 11th, 2013 that we need to leave by January 31st, 2014. The moment we received the letter we began looking and found a house to rent. We were told of our approval of our application last week. We sign the lease tomorrow and were told we can move in on the first of January. We are aware of the lease and giving the landlord a 30 day notice of our leave, but figured that since they are already vacating the entire building they wouldn't need a notice. We will be gone by the 1st of January. We will write a letter to landlord telling them we are leaving, but are we still obligated to pay rent for the month of January?